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\title{Faraday Rotation Measure Synthesis for Magnetic Fields of Galaxies}

\author[P.~Frick et al.]
       {P.~Frick$^1$, D.~Sokoloff$\,^{2}$,  R.~Stepanov$^1$,
        and R.~Beck$^3$\\
$^1$ Institute of Continuous Media Mechanics,
Korolyov str.~1, 614013, Perm, Russia \\
$^2$ Department of Physics, Moscow University, 
119899, Moscow, Russia \\
$^3$ Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel 69,
  53121 Bonn, Germany}

%\begin{document}

\date{Accepted 2010 .... Received 2010 ....; in original form 2010}

\pagerange{\pageref{firstpage}--\pageref{lastpage}}
\pubyear{2009}
\begin{document}
\maketitle

\label{firstpage}

%
%******************************************************************************
\begin{abstract}

RM Synthesis was recently elaborated as a new tool for the
interpretation of polarized emission data in order to separate the
contributions of different sources lying on the same line of sight.
Until now the method was mainly applied to discrete sources in
Faraday space (Faraday screens). Here we consider how to apply the
RM Synthesis to extract the information concerning the magnetic
fields of extended sources, e.g. galaxies. The main attention is
given to two related novelties in the method, i.e. the symmetry
argument in Faraday space and the wavelet technique.

We give a relation between our method and the previous applications
of RM Synthesis to point-like sources. We demonstrate that the
traditional RM Synthesis for a point-like source indirectly exploits
a symmetry argument and, in this sense, can be considered as a
particular case of the method presented here. Investigating the
applications of RM Synthesis to polarization details associated with
small-scale magnetic fields, we isolate an option which was not
covered by the ideas of the Burn theory. We describe the
contribution of small-scale fields in terms of Faraday dispersion
and beam depolarization. We exploit the complex polarization for RM
Synthesis without any averaging and demonstrate that it allows to
obtain a range in Faraday space where the contribution from
small-scale field is located.

A general conclusion concerning the applicability of RM Synthesis to
the interpretation of the radio polarization data for extended
sources, like spiral galaxies, is that quite severe requirements are
needed to exploit the method in full extent. If the wavelength range
of observations is not adequate we can pretend to reconstruct only
some features of the Faraday spectrum.

\end{abstract}



\begin{keywords}
Methods: polarization -- methods: data analysis -- galaxies:
magnetic fields -- RM Synthesis -- wavelets
\end{keywords}
% --------------------------------------------------------------------

\section{Introduction}
\label{intro}

RM Synthesis is a new tool for the interpretation of polarized
emission data in order to get information on the emitting media
\citep{2005A&A...441.1217B}. Until now the method was mainly applied
to discrete sources in Faraday space (``Faraday screens'')
\citep{heald2009}. The aim of this paper is to discuss applications
to extended sources.

We remind that the idea of RM Synthesis is based on the fact that
the complex polarized intensity $P$ can be calculated as a
Fourier transform
%
\begin{equation}
\label{p_to_f} P(\lambda^2) =  \int_{0}^{\infty} F(\phi) e^{2i\phi
\lambda^2}  d \phi
\end{equation}
%
of the Faraday dispersion function $F$ in Faraday space with the
coordinate $\phi$ which is the Faraday depth
\citep{Burn1966MNRAS.133...67B}. Performing the inverse Fourier
transform of $P$ one obtains $F$ which is the polarized intensity
emerging from a region with Faraday depth $\phi$. Faraday depth is
defined as
%
\begin{equation}
\phi(z) = 0.81\int_{0}^{z} B_\parallel
(z) n_e(z) dz', \label{fardep}
\end{equation}
%
where $B_\parallel$ is the line-of-sight magnetic field component
measured in $\mu$G, $n_e$ is the thermal electron density measured
in cm$^{-3}$ and the integral is taken from the observer (who is
supposed to be at $z=0$) to the current point $z$ along the line of
sight over the region which contains both, magnetic fields and free
electrons, where $z$ is measured in parsecs.

In the context of RM Synthesis, one has to make a distinction
between Faraday depth and Faraday rotation measure $RM$ which is
defined as
%
\begin{equation}
RM (\lambda) ={{d  \Psi} \over {d (\lambda^2)}},
\label{RM}
\end{equation}
%
where $\Psi$ is the polarization angle ($P = |P| e^ {2 i \Psi}$).

The particular topics addressed in this paper are as follows. First,
we formulate the RM Synthesis problem in terms of wavelet transform
as suggested by \cite{2010MNRAS.401L..24F} (Section \ref{wavelet}).
We appreciate that the available bulk of results on RM Synthesis was
addressed mainly to point sources in Faraday space and exploit
concepts as the RM Spread Function and the RM cleaning procedure. In
Sections \ref{rmsf} and \ref{clean} we demonstrate how these
concepts can be presented in the framework of wavelet approach. We
note that both concepts are not specific for the wavelet approach.
Correspondingly, we conclude that the results obtained in our
approach can be reduced to the traditional ones after appropriate
rescaling.

In Section \ref{galactic} we discuss what kind of Faraday dispersion
function can be expected from galaxies considered as extended
sources of polarized emission in Faraday space. Of course this shape
differs from that one expected from a point source. We demonstrate
that symmetric distributions of $F(\phi)$ appear to be typical for
galactic discs.

The efficiency of RM Synthesis is crucially determined by the
expected range of Faraday depths typical for the source of interest
and the spectral range $\lambda_{\rm min} < \lambda < \lambda_{\rm
max}$ covered by observations. We analyze the efficiency of the new
generation of radio telescopes (LOFAR and its combinations with
other devices) in context of recognition of extended (galaxy-like)
objects (Section \ref{range}). In general, the lower the Faraday
depth and the lower the minimal wavelength $\lambda_{\rm min}$, the
more efficient RM Synthesis becomes. For the sake of definiteness,
we consider as a typical example the observational range in the
LOFAR high band ($1.25 < \lambda < 2.5$\, m) and use another
observational range if required to illustrate particular properties
of RM Synthesis.

Another factor which determines the efficiency of RM Synthesis is
the spectral coverage by the frequency canals of the performed
observations. We demonstrate in Section \ref{sampling} that for poor
sampling an equidistant spacing in $\lambda^2$ space is preferable
in this respect. In Section \ref{turb} we analyze the contribution
of small-scale (turbulent) fields to the Faraday dispersion function
and show that RM Synthesis can recognize them even if structures in
global galactic scales remain invisible because the $\lambda_{\rm
min}$ for the observational range is too large.

In Section~\ref{SE} we present a synthetic example which brings
together various aspects of RM Synthesis presented and finally we discuss
in Section~\ref{DC}
general perspectives of RM Synthesis in context of the wavelet-based
algorithm.


\section{Wavelet-based RM Synthesis }
\label{wavelet}

The inversion of formula (\ref{p_to_f}) shows that the Faraday
dispersion function $F$ is the Fourier transform of the complex
polarized intensity:
\begin{equation}
\label{f_to_p1}
F(\phi) = {{1} \over{\pi}} \hat P(k),
\label{Burn}
\end{equation}
where $k=2\phi$, and the Fourier transform is defined as
\begin{equation}
\label{four}
\hat {f}\left( k \right) = \int_{-\infty}^{\infty} {f\left( x \right)e^{ - i k x} dx}.
\end{equation}
The practical limitation of the use of Eq.~(\ref{f_to_p1}) comes
from the fact that $P$ is defined only for $\lambda^2>0$ and in
practice can be observed only in a finite spectral band. Moreover,
the maximum of $P$ is often located outside the available spectral
window $\lambda_{\rm min}<\lambda<\lambda_{\rm max}$.

The lower bound $\lambda_{\rm min}$ restricts the possibility to
recognize objects which are extended in Faraday space, the upper
bound $\lambda_{\rm max}$ suppresses the visibility of small-scale
structures of the object in Faraday space and the lack of negative
$\lambda^2$ impedes the correct reconstruction of the intrinsic
polarization angle. Large gaps in wavelength sampling inside
the observations window prevent the reconstruction of objects of
large Faraday depths. All these problems and limitations can also be
illustrated using the wavelet representation, which provides a kind
of local Fourier transform, isolating a given structure in physical
space and in Fourier space.

The wavelet transform of the Faraday dispersion function $F(\phi)$
can be written in the form
%
\begin{equation}
\label{wF_d}
w_F(a,b) = {{1}\over {|a|}} \int\limits_{ - \infty }^\infty
{F(\phi)\psi ^\ast \left( {\frac{\phi - b}{a}} \right)d\phi} ,
\end{equation}
%
where $\psi(\phi)$ is the analyzing wavelet, $a$ defines the scale
and $b$ defines the position of the wavelet center.
Then the coefficient $w_F$ gives the contribution of the
corresponding structure to the function $F$.

From the relation between the Faraday function and polarized
emission (Eq.~\ref{Burn}) and the definition of the wavelet
transform (Eq.~\ref{wF_d}) one can get the wavelet decomposition of
the Faraday dispersion function from the complex polarized intensity
$P(\lambda^2)$
%
\begin{equation}
\label{wF_P} w_F(a,b) = {{1}\over {\pi}} \int\limits_{ -
\infty}^\infty {P(\lambda^2)e^{-2ib \lambda^2} \hat{\psi}^\ast
\left( -2a \lambda^2 \right)d \lambda^2} .
\end{equation}
%
Then the function $F$ can be restored using the inverse transform
%
\begin{equation}
\label{wF_inv}
F(\phi) = \frac{1}{C_\psi }\int\limits_{-\infty}^\infty {\int\limits_{ -
\infty }^\infty {\psi \left( {\frac{\phi - b}{a}} \right)w_F\left(
{a,b} \right)\frac{d a \, d b}{a^2}} }.
\end{equation}
%
The reconstruction formula (\ref{wF_inv}) exists under the condition
that
%
 \begin{equation}
\label{adm}
C_\psi = \frac1{2}\int\limits_{ - \infty }^\infty
{\frac{\vert \hat {\psi }(k )\vert ^2}{ |k| }d k < \infty } .
\end{equation}
%
Here $\hat{\psi }(k ) = \int {\psi (\phi)e^{ - ik\phi}d\phi} $ is
the Fourier transform of the analyzing wavelet $\psi(\phi)$. We will
use below the so-called Mexican hat $\psi (\phi) = (1-\phi^2) \exp
(-\phi^2/2)$ as the analyzing wavelet. The wavelet is real, however,
the function $P$ is complex, so the wavelet coefficients $w_F$
are complex as well. For the chosen wavelet $w_F(-a,b)=w_F(a,b)$ and
$C_\psi = 1$.

\cite{2010MNRAS.401L..24F} suggested to use the symmetry of the
reconstructed object to restore $P(\lambda^2)$ in the domain of
negative $\lambda^2$. In the wavelet-based algorithm they
generalized this method of $P(\lambda^2)$ extension for the case of
a sample of isolated objects in Faraday space. First, they divided
Eq.~(\ref{wF_P}) in two parts $w_F(a,b) =w_-(a,b)+w_+(a,b)$, where
\begin{eqnarray}
\label{wF_P2} w_-(a,b)&=& {{1}\over {\pi}}
\int\limits_{ - \infty }^0
{P(\lambda^2)e^{-2ib \lambda^2} \hat{\psi}^\ast \left( -2a
\lambda^2 \right)d\lambda^2},  \\   w_+(a,b)&=&{{1}\over {\pi}} \int\limits_0^\infty
{P(\lambda^2)e^{-2ib \lambda^2} \hat{\psi}^\ast \left( -2a \lambda^2 \right)d\lambda^2}.
\end{eqnarray}

Secondly, \cite{2010MNRAS.401L..24F} suggested to calculate the
coefficients $w_+(a,b)$ from the known $P(\lambda^2)$ for
$\lambda^2>0$ and to recognize the dominating structures in the map
$|w_+(a,b)|$. The coordinate $b$ of the corresponding maximum gives
the value of $\phi_0^i$, where the upper index $i$ indicates the
number of the structure. Then the coefficients $w_-(a,b)$ can be
reconstructed, following the symmetry arguments (see
Eq.~(\ref{cont}) below). If several maxima are available, the
algorithm is applied for the local domain in wavelet space $(a,b)$
and defined
\begin{equation}
\label{w-w+}
w_-(a,b)=w_+\left(a,2\phi_0^i(a,b)-b\right),
\end{equation}
where the parameter $\phi_0^i(a,b)$ for the given point $(a,b)$ was
chosen according to the structure $i$ which dominates for this point.


\section{Rotation Measure Spread Function }
\label{rmsf}

The reconstructed Faraday dispersion function $\tilde F(\phi)$ can
be related to the true Faraday dispersion function $F(\phi)$ by
$R(\phi)$, the ``RM Spread Function'' (RMSF) \footnote{The correct
name should be ``Faraday depth dispersion function'', but the
original name is kept here for the sake of consistency.}
\begin{equation}
\tilde F(\phi) = F(\phi)\ast R(\phi),
\label{rmsf0}
\end{equation}
where $\ast$ denotes the convolution. The RMSF is defined as
\begin{equation}
R(\phi) = K \int_{-\infty}^{\infty} W(\lambda^2)e^{-2i\phi \lambda^2}d \lambda^2 ,
\label{rmsf1}
\end{equation}
where $W(\lambda^2)$ is the shape of the observable window in
$\lambda^2$ space (the window function, which can include weights
due to different signal-to-noise ratios in the observation
channels, see \cite{2009IAUS..259..591H}) and $K$ is a
normalization constant.

\begin{figure}
\begin{picture}(240,330)(0,0)
\put(0,220){\includegraphics[width=0.4\textwidth]{figR.eps}}
\put(0,110){\includegraphics[width=0.4\textwidth]{figRBB.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{figRW.eps}}
\put(210,320){(a)}
\put(210,210){(b)}
\put(210,100){(c)}
\end{picture}
\caption {RM spread functions for a single observation window with
$1.25<\lambda<2.5$\,m: (a) standard $R$, (b) $R_{BB}$ and (c)
$R_W$.  Real part (thin solid red), imaginary part (dashed blue),
modulus (thick black).  } \label{fig-rmsf}
\end{figure}

For a single spectral window ($W=1$  for $\lambda_{\rm min} <
\lambda < \lambda_{\rm max}$ and W=0 elsewhere) the RMSF has the
simple form
\begin{equation}
R(\phi) = K e^{-2i\phi \lambda^2_0} \, \frac{\sin{(\phi
\Delta\lambda^2)}}{\phi}, \label{rmsf2}
\end{equation}
where $\lambda_0^2 = (\lambda_{\rm \min}^2 + \lambda_{\rm max}^2)/2$
and $\Delta \lambda^2 = (\lambda_{\rm max}^2 - \lambda_{\rm
min}^2)/2$. $R(\phi)$ is shown in Fig.~\ref{fig-rmsf}a for
$\lambda^2_0=3.91$m$^2$ and $\Delta\lambda^2=2.35$m$^2$
(corresponding to the LOFAR wavelength range in the highband).
Formula (\ref{rmsf2}) shows that the RMSF in this case is a
complex-valued function which means that the reconstructed Faraday
function differs from the true one not only in amplitude, but also
in phase (due the lack of data in the domain of negative $\lambda^2$
RM Synthesis rotates the function under reconstruction). To avoid
this shortcoming \cite{2005A&A...441.1217B} proposed to multiply
Eq.~(\ref{rmsf2}) by the factor $e^{2i\phi \lambda^2_0}$ which means
that
\begin{equation}
R_{BB}(\phi) = K \int_{-\infty}^{\infty} W(\lambda^2)e^{-2i\phi
(\lambda^2-\lambda^2_0)}d \lambda^2  = K\frac{\sin(\phi
\Delta\lambda^2)}{\phi} \label{rmsfBB}\end{equation} and
\begin{equation}
\tilde F(\phi) = e^{-2i\phi \lambda^2_0}F(\phi)\ast R_{BB}(\phi).
\label{rmsf01}
\end{equation}
Thus, the modified spread function $R_{BB}(\phi)$ has a vanishing
imaginary part and a real part, which reproduces the shape of the
envelope of the function $R(\phi)$ (Fig.~\ref{fig-rmsf}b).

Another way to improve the spread function follows from
\cite{2010MNRAS.401L..24F}. Their algorithm uses the symmetry
argument (the results of Sect.~5 give strong support for this
argument - realistic objects mainly look like even objects in
Faraday depth space). It means that if the object is centered at
Faraday depth $\phi_0$, then $F(2\phi_0-\phi)=F(\phi)$. In
$\lambda^2$ space this gives
%
\begin{equation}
P(-\lambda^2) = \exp{-4 i \phi_0 \lambda^2} P(\lambda^2),
\label{cont}
\end{equation}
which means in terms of the spread function for a single window
\begin{equation}
R_{W}(\phi) =  2K\frac{\sin{(\phi \Delta\lambda^2)}}{\phi} \cos{(2\phi \lambda^2_0)}
\label{rmsfW}\end{equation}
and
\begin{equation}
\tilde F(\phi) = e^{-4i\phi_0\lambda^2_0}F(\phi)\ast R_{W}(\phi).
\label{rmsf00}
\end{equation}
The spread function $R_W(\phi)$ is also shown in
Fig.~\ref{fig-rmsf}c. Similar to $R_{BB}$ this function has no
imaginary part, but the real part remains exactly the same as in
Eq.~(\ref{rmsf2}).


\section{Cleaning and wavelets}
\label{clean}

We discussed above the recognition of extended objects in Faraday
space with an internal structure characterized by some symmetry. The
simplest objects for recognition are point-like sources, which can be
described in Faraday space by delta functions
\begin{equation}
F(\phi)=F_0 e^{2i\chi_0}\delta(\phi-\phi_0).
\label{point_source}
\end{equation}
The corresponding polarized intensity is given by
\begin{equation}
P(\lambda^2)=F_0 e^{2i\chi_0}e^{2i\phi_0\lambda^2}.
\label{point_P}
\end{equation}
RM Synthesis has to recognize point-like sources located on the same
line of sight. This problem can be solved using a deconvolution
procedure using the Faraday dispersion function, called ``RM-CLEAN''
\citep{heald2009}. This procedure first recognizes the maximum in
the reconstructed Faraday dispersion function and then iteratively
subtracts the scaled versions of the RMSF until the noise level is
reached, after which a smoothed representation of the ``CLEAN
model'' is used as the approximate true Faraday dispersion function.
After removing the brightest structure (including the corresponding
sidelobes) the procedure finds the maximum in the remaining data and
restarts the iteration. The efficiency of this cleaning procedure
was demonstrated in \cite{heald2009}, where it was used for the data
cubes of the Westerbork SINGS survey.

\begin{figure}
\begin{picture}(240,330)(0,0)
\put(0,220){\includegraphics[width=0.4\textwidth]{dataCLEAN.eps}}
\put(0,110){\includegraphics[width=0.4\textwidth]{heald1.eps}}
\put(5,0){\includegraphics[width=0.4\textwidth]{heald2.eps}}
\put(210,320){(a)}
\put(210,210){(b)}
\put(210,100){(c)}
\end{picture}
\caption {Example of the RM-CLEAN process from
\protect\cite{heald2009} for a bright point source in the field of
the galaxy NGC~7331.  Top: observed $Q$ (black) and $U$ (non black);
middle: the restored $F(\phi)$ before cleaning (black) and after
cleaning (non black); bottom: observed polarization angles (black)
and reconstructed after cleaning (non black).} \label{fig_clean}
\end{figure}
\begin{figure}
\begin{picture}(240,335)(0,0)
\put(1,225){\includegraphics[width=0.4\textwidth]{Wclean1.eps}}
\put(1,110){\includegraphics[width=0.4\textwidth]{Wclean2.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{Wclean_res.eps}}
\put(210,320){(a)}
\put(210,210){(b)}
\put(210,100){(c)}
\end{picture}
\caption {Wavelet cleaning for the same example as in
Fig.~\ref{fig_clean}. (a) wavelet plane after the first step (the
symmetry of the main maximum is used); (b) wavelet plane after
cleaning of the first point source and using symmetry argument for
the rest (note that the amplitudes in the two figures are
different); (c) observed polarization angles (black) and
reconstructed after two-step wavelet cleaning (non black). Here and
below colors are given in a relative scale specific for each panel
and black color corresponds to maximum.} \label{fig_clean2}
\end{figure}

Let us show how this cleaning can be realized in terms of
wavelet-based RM Synthesis. We exploit the example, presented in
Fig.~3 from \cite{heald2009}. This figure is partially reproduced in
our Fig.~\ref{fig_clean}, where the observed $U(\lambda^2)$ and
$Q(\lambda^2)$, the restored $F(\phi)$ before and after cleaning and
the polarization angles observed and reconstructed after cleaning
are shown. The observations are done in two windows. In
Fig.~\ref{fig_clean2}a we show the modulus of the wavelet
coefficients $|w_{+}|$, calculated from same observed $U(\lambda^2)$
and $Q(\lambda^2)$. Then using the symmetry argument for the main
maximum we find $|w_{-}|$ and reconstruct $F(\phi)$. Taking the
dominating maximum in $F(\phi)$ at $\phi=-185$ rad m$^{-2}$ we
replace it by an equivalent point source and calculate for this
point source the wavelet transform $w_1$, based on the same sample
of observational channels as in original data. Then we subtract
$w_1$ from the wavelet transform $w$ of original data and we obtain
the wavelet image, shown in the second panel of
Fig.~\ref{fig_clean2}b, which demonstrates a dominating maximum at
$\phi=165$ rad m$^{-2}$. Using again the symmetry argument we get
$w_{-}$ and restore the second point-like source in $F(\phi)$. The
corresponding polarization angles, calculated for the given
observational windows are shown in the panel (c), which is
practically equivalent to the result of RM-CLEAN
(Fig.~\ref{fig_clean}). Summarizing this section, we conclude that,
in contrast to the extended sources, for a source that is point-like
in Faraday space, wavelet-based RM Synthesis gives just the same
result as conventional RM Synthesis.


\section{Galactic magnetic field in Faraday depth space}
\label{galactic}

\begin{figure}
\begin{picture}(240,220)(0,0)
\put(0,110){\includegraphics[width=0.4\textwidth]{fig1a.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{fig1b.eps}}
\put(210,210){(a)}
\put(210,100){(b)}
\end{picture}
\caption {Typical shapes of magnetic field distribution (a):
$B_1$ (solid line, red), $B_2$ (dashed, blue), $B_3$ (dot-dashed, black) and
 (b) corresponding Faraday depths $\phi$.  }
\label{fig1}
\end{figure}

The complex-valued intensity of polarized radio emission for a given wavelength
%
\begin{equation}
\label{p-def} P(\lambda^2) =  \int_{0}^{\infty}
\varepsilon(z)e^{2i\chi(z)} e^{2i\phi(z) \lambda^2}  d z
\end{equation}
%
is defined by the emissivity $\varepsilon$ and the intrinsic
polarization angle $\chi$ along the line of sight. The integral is
taken over the whole emitting region. The emissivity depends on the
relativistic electron density $n_c$, the magnetic field component
perpendicular to the line of sight and the synchrotron spectral
index $\alpha$ ($\alpha = 0.9$, see below)
%
\begin{equation}
\label{vsreps} \varepsilon(z) = n_c(z) |B_\perp(z) |^{1+\alpha} \, .
\end{equation}
%

\begin{figure}
\begin{picture}(240,220)(0,0)
\put(1,110){\includegraphics[width=0.4\textwidth]{fig2a.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{fig2b.eps}}
\put(210,210){(a)}
\put(210,100){(b)}
\end{picture}
\caption{Real part of the Faraday dispersion function $F(\phi)$
(panel a), ${\rm Im} \, F=0$, and the modulus of corresponding
$P(\lambda^2)$ (panel b) for different magnetic field profiles:
$B_1$ (solid line, red), $B_2$ (dashed, blue), $B_3$ (dot-dashed,
black). }
 \label{fig2}
\end{figure}

Our analysis starts from an investigation what are typical shapes of
the Faraday dispersion functions expected for the line of sight
crossing galactic discs with several simple distributions of the
parallel magnetic field, which provides the Faraday rotation. These
distributions are shown in Fig.~\ref{fig1}a. We consider two
symmetric distributions with respect to center at $z=z_0$. The first
one has a sharp boundary at $z=z_0\pm h$ \begin{eqnarray}
\label{dis1}
 B_{1\parallel}(z) =
\left\{
  \begin{array}{ll}
   a_1, & |z-z_0|\leq h \\
   0, & |z-z_0|>h
  \end{array}.
\right.
\end{eqnarray}
Here $z_0$ is position of galactic equatorial plane and $h$ is the
half thickness of the galactic disc. The second distribution has a
smooth shape approximated by a Gaussian function with a
characteristic half thickness $h$
\begin{equation}\label{dis2}
    B_{2\parallel}(z)=a_2 e^{ -(z-z_0)^2/ h^2}.
\end{equation}

\begin{figure}\centering{
\includegraphics[width=0.4\textwidth]{fig2c.eps}}
\caption {Faraday dispersion function $F(\phi)$ for different
separations $\Delta z$ between the regions of synchrotron emission
and Faraday rotation with Gaussian profiles: $\Delta z =0.1$ kpc
(solid line, red), $\Delta z=0.2$ kpc (dashed, blue), $\Delta z=0.5$ kpc
(dot-dashed, black).}
 \label{fig2a}
\end{figure}

For comparison we take into consideration a distribution with a
field reversal along the line of sight
\begin{equation}\label{dis3}
    B_{3\parallel}(z)=-a_3{{(z-z_0)} \over {h}} e^{ -(z-z_0)^2/ h^2}.
\end{equation}
We choose an amplitude of magnetic field $a_1=2\,\mu{\rm G}$ and
$h=0.5\,{\rm kpc}$. The densities of relativistic electrons $n_c$
and thermal electrons $n_e$ measured in cm$^{-3}$ are supposed to
have Gaussian profiles
\begin{equation}\label{dise}
    n_c(z)= Ce^{ -(z-z_0)^2/ h^2}, \hspace{1cm} n_e(z)= 0.03\, e^{ -(z-z_0)^2/ h^2}.
\end{equation}
Here $C$ (measured in cm$^{-3}$) is the relativistic electron
density at the galactic midplane. We use $C$ as a normalization
constant for $F$ and $P$, so that they are numerically evaluated in
arbitrary but mutually consistent units. The amplitudes $a_2$ and
$a_3$ are adjusted to obtain the same maximal Faraday depth for all
three distributions. Fig.~\ref{fig1}b shows that the symmetric
profiles of the parallel magnetic field (first two cases) lead to a
monotonic increase (decrease) of Faraday depth along the coordinate
$z$ inside the galaxy, while the antisymmetric profile (last case)
leads to some inversion - the central galactic plane becomes the
most remote (or most nearby) point in Faraday depth space.


\begin{figure}
\begin{picture}(240,460)(0,0)
\put(0,345){\includegraphics[width=0.40\textwidth]{fig6a.eps}}
\put(0,230){\includegraphics[width=0.40\textwidth]{fig6b.eps}}
\put(0,115){\includegraphics[width=0.40\textwidth]{fig6c.eps}}
\put(0,000){\includegraphics[width=0.40\textwidth]{fig6d.eps}}
\put(210,445){(a)}
\put(210,330){(b)}
\put(210,215){(c)}
\put(210,100){(d)}
\end{picture}
\caption {Wavelet plane  $|w_F(a,b)|$  for various spectral windows.
Panel (a) $0.06<\lambda<2.5$\, m; (b) $0.21<\lambda<2.5$\, m;
(c) $1.25<\lambda<2.5$\, m; (d) $0.06<\lambda<0.21$\,m.  } \label{fig6}
\end{figure}

For the sake of normalization we choose the direction of the
perpendicular field to obtain a purely real $P$. We consider the scales
$h_c=2h_e=2h$. The perpendicular component is described by the same
distributions as for the parallel component in
Eqs.~(\ref{dis1})-(\ref{dis3}). Namely, we take
$B_{1\perp}=B_{1\parallel}$, $B_{2\perp}=B_{2\parallel}$ and
$B_{3\perp}=B_{2\parallel}$.

Fig.~\ref{fig2}a shows that both even profiles ($B_1$ and $B_2$)
produce in Faraday space basically box-like distributions with sharp
boundaries. In contrast, Faraday depth in case of the magnetic field
with a reversal varies slowly near the point where Faraday depth
becomes minimal and has a sharp point-like structure in $F(\phi)$
near the point where $\phi$ is maximal while $F$ is still
nonvanishing.


\begin{figure}
\begin{picture}(240,220)(0,0)
\put(0,110){\includegraphics[width=0.4\textwidth]{fig66a.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{fig66b.eps}}
\put(210,210){(a)}
\put(210,100){(b)}
\end{picture}
\caption {Test Faraday dispersion function (black dots in both
panels) and results of RM Synthesis performed for various spectral
windows. Panel (a): $0.06<\lambda<2.5$\, m (solid, red line),
$0.21<\lambda<2.5$\, m (dashed, blue), $1.25<\lambda<2.5$\, m
(dash-dot, black). Panel (b): $0.06<\lambda<1.0$\, m (solid, red
line), $0.06<\lambda<0.42$\, m (dashed, blue), $0.06<\lambda<0.21$\,
m (dash-dot, black).} \label{fig66}
\end{figure}

We conclude that box-like and point-like shapes of $F(\phi)$ are
quite typical for disc components of spiral galaxies. Of course, one
can obtain more smooth $F(\phi)$ by taking wider distributions for
thermal electrons than for cosmic rays. According to
Fig.~\ref{fig2}b the level of polarized intensity at large
$\lambda^2$ is higher for point-like $F$ than for box-like ones.
\footnote{But there is no point-like F in Fig.5a! Do you mean the
dot-dashed F? Furthermore, is the larger level of P good or bad for
RM Synthesis? Please clarify. --Rainer}

In general, the synchrotron emission is not necessarily located in
the same region as the Faraday rotation. This leads to a so-called
``Faraday screen'' which can be due to a magnetic field
configuration \footnote{Unclear: Which sort of configuration can
produce a Faraday screen? --Rainer} or to a difference between
distributions of $n_c(z)$ and $n_e(z)$. For simplicity, we shift
$\varepsilon(z)$ by some distance $\Delta z$. The resulting Faraday
dispersion functions for different Faraday screen separations are
shown in Fig.~\ref{fig2a}. For larger $\Delta z$ the Faraday
dispersion function loses symmetry and looks more like a point
source. \footnote{But there no symmetry in any of the curves in
Fig.6! Please clarify. --Rainer.}


\section{The role of the spectral range}
\label{range}

The efficiency of RM Synthesis crucially depends on the
observational range $\lambda_{\rm min} < \lambda < \lambda_{\rm
max}$. Wavelets help to understand the role of these parameters.
Fig.~\ref{fig6} shows the wavelet planes for the first artificial
example (solid lines in Fig.~\ref{fig2}), calculated for different
observational windows. Each panel shows the absolute value of the
wavelet coefficient $W(a,b)$ in the $(a,b)$ plane. The horizontal
axis ($b$) presents the position in Faraday depth space, the
vertical axis gives the scale $a$ in logarithmic presentation. The
upper bound is first fixed at $\lambda_{\rm max}=2.5$\,m. The panel
(a) shows the wavelet plane simulated for the lower bound
$\lambda_{\rm min}=6$\,cm (thus $0.0036<\lambda^2<6.25$\, m$^2$).
The large horizontal dark feature at $a \approx 40$\, rad m$^{-2}$
corresponds to the box-like structure, and the two tapering
structures are generated by the sharp borders of this box. The next
panel shows what remains in the wavelet plane if the lower bound is
moved to $\lambda_{\rm min}=21$\,cm (the window
$0.044<\lambda^2<6.25$\, m$^2$). The central spot (responsible for
the box) is almost completely lost even at these relatively short
wavelengths. Applying the actual LOFAR window
($1.56<\lambda^2<6.25$\, m$^2$), only weak traces of side horns
remain in the wavelet plane (panel c). The lower panel (d) in
Fig.\ref{fig6} shows the wavelet plane for a relatively
high-frequency window ($0.06<\lambda<0.21$\, m), which perfectly
keeps all information concerning the large scales, but completely
loses any information concerning the small scales, responsible in
this example for the abrupt boundaries.

Fig.~\ref{fig66} shows how the shrinking of the wavelength range of
observations reduces the quality of the Faraday function
reconstructed from the corresponding sets of $w_F(a,b)$. We see that
increasing of $\lambda_{\rm min}$ leads to a graduate decay of the
reconstructed $F$ in the main bulk of the structure under
reconstruction (Fig.~\ref{fig66}a). Starting from $\lambda_{\rm
min} = 21$ cm we reconstruct details associated with the boundaries
only. Diminishing the upper bound $\lambda_{\rm max}$ we lose the
small-scale details, restoring a smooth structure only
(Fig.~\ref{fig66}b).

The possibility to reconstruct a structure of scale $\Delta \phi$ in
Faraday space is determined by the quantity $X= \Delta \phi \,
\lambda$ where $\lambda$ is the wavelength at which the observations
are performed. We illustrate the role of this quantity for the
large-scale structure from Fig.~\ref{fig66}a. The relevant
parameters here are $\Delta \phi = 38$ rad m$^{-2}$ and $\lambda =
\lambda_{\rm min}$. $X=1.68$ is for the dashed (blue) line and
$X=60$ for the dash-dot (black) lines. Because the latter curve
keeps the information of sharp ends of the distribution $F(\phi)$
and even the first curve looses a substantial part of the signal, we
conclude that
%
\begin{equation}
X = \Delta \phi \, \lambda_{\rm min}^2 \lesssim 1
 \label{crit}
\end{equation}
%
is the condition to observe a galaxy as an extended source in
Faraday space rather than as a couple of point-like sources. For
observations in the LOFAR highband ($\lambda_{\rm min} = 1.25$\,m)
this gives $\Delta \phi = 0.64$\,rad m$^{-2}$, which is at least one
order of magnitude lower than a typical value for contemporary
galaxies. If $\lambda_{\rm min} =0.167$\,m (by including
observations e.g. with the ASKAP telescope) this yields a much more
comfortable $\Delta \phi^*=35$\,rad m$^{-2}$.

For distant galaxies the wavelength in the framework of the source
is $\lambda/(1+Z)$ ($Z$ is the redshift) and Eq.~(\ref{crit}) reads
%
\begin{equation}
X = \Delta \phi \, \lambda_{\rm min}^2 /(1 +Z)^2\lesssim 1.
 \label{zcrit}
\end{equation}
Taking $Z=3$ one obtains that $\Delta \phi \approx 10$\,rad m$^{-2}$
for LOFAR-type observations. LOFAR observations can also be
addressed to isolate a contribution of weak intergalactic magnetic
fields, which can give typical $\Delta \phi$ much lower than that
for galactic magnetic fields.

Note that we used RM Synthesis to reconstruct the Faraday dispersion
function $F$ in Fig.~\ref{fig6}a and exploit wavelets in two lower
panels for interpretation of the results obtained only. We could
apply the wavelet-based algorithm of RM Synthesis for the range
$0.06<\lambda<2.5$\, m (e.g. by combining observations from the EVLA
and LOFAR telescopes) and obtain a more adequate reconstruction
(solid line in Fig.~\ref{fig66}a) based on the symmetry arguments as
suggested by \cite{2010MNRAS.401L..24F}, see also Sect.~2. The
observational ranges ($0.21<\lambda<2.5$\, m, $1.25<\lambda<2.5$\,
m) appear to be ***useless*** for this purpose, because the maximum
in the wavelet plane is lost in the panels (b) and (c) of
Fig.~\ref{fig6}.

\begin{figure}
\begin{picture}(240,220)(0,0)
\put(0,110){\includegraphics[width=0.4\textwidth]{fig_sampling1.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{fig_sampling2.eps}}
\put(210,210){(a)}
\put(210,100){(b)}
\end{picture}
\caption {Results of wavelet-based RM Synthesis performed for the
observational window $0.06<\lambda<2.5$\, m  with 1024 (a)
and 128 (b) channels providing uniform sampling in
$\lambda^2$ space (dashed black line) or uniform sampling in
frequency space (solid red line). } \label{fig_sampling}
\end{figure}



\section{The role of wavelength spacing}
\label{sampling}

As we conclude above, RM Synthesis of extended magnetic
configurations requires a proper wavelength range of observations.
$\lambda_{\rm min}$ has to be small enough to reproduce the grand
design of $F$ and a large $\lambda_{\rm max}$ is required to
reproduce sharp details of $F$. Obviously, RM Synthesis demands a
substantial number of channels in the observational range to perform
the Fourier transform included in the method. ***A dispersion
function $F$ which is extended over a large range of Faraday depths
requires a large number of channels. If the number of channels is
limited, the main question is how to distribute the channels***
between the given values of $\lambda_{\rm min}$ and $\lambda_{\rm
max}$. We clarify this using the example exploited for
Fig.~\ref{fig6}. For the sake of the definiteness we compare two
samplings, i.e. a uniform spacing in frequency  space and that one
in the $\lambda^2$ space.

We presume that the observational range is adequate for application
to the example \footnote{Unclear: adequate in which respect?
--Rainer} and use $\lambda_{\rm min} = 6$\,cm and $\lambda_{\rm max}
= 2.5$\,m and apply the wavelet-based RM Synthesis.
Fig.~\ref{fig_sampling}a presents the results for a large (1024)
number of channels. We see that even here a uniform spacing in the
$\lambda^2$ looks better than that one in frequency space, which
gives a more noisy result. The difference between the two kinds of
spacings becomes much more pronounced for a moderate (128) number of
channels. The results for the uniform spacing in $\lambda^2$ is more
or less similar to that one in the upper panel, while for the
uniform spacing in frequency the noise in $F$ becomes comparable to
the signal. It is clear that the effect is essential for a large
windows (namely, for large ratio $\lambda_{\rm max}/\lambda_{\rm
min}$) because a uniform spacing in frequency leads to points
crowding at small $\lambda^2$. The poor sampling at large
$\lambda^2$ (responsible for small scales in Faraday depth space)
results in small-scale noise in the reconstructed signal. Note that
the advantages of the uniform spacing in $\lambda^2$ was discussed
by \cite{1979A&A....78....1R}. \footnote{Please add why this paper
is important to mention here. --Rainer}

\begin{figure}
\begin{picture}(240,225)(0,0)
\put(0,115){\includegraphics[width=0.4\textwidth]{fig7a.eps}}
\put(0,00){\includegraphics[width=0.4\textwidth]{fig7b.eps}}
\put(210,210){(a)}
\put(210,100){(b)}
\end{picture}
\caption {Magnetic field distributions over a galactic disc. The
ratio of small-scale ($b$) and large-scale ($B$) components is
$b/B=2$, the ratio of spatial scales of the components is $l/2h=
0.1$. The upper panel presents the light-of-sight magnetic field and
the lower panel gives the polarized intensity. The long and short
dashed lines give $P$ for two different realizations of the
turbulent magnetic fields calculated for a very narrow beam, while
the solid line shows $P$ calculated for a beam which contains 50
independent turbulent cells.} \label{fig7}
\end{figure}


\begin{figure}
\begin{picture}(240,345)(0,0)
\put(0,230){\includegraphics[width=0.40\textwidth]{fig9a.eps}}
\put(0,115){\includegraphics[width=0.40\textwidth]{fig9b.eps}}
\put(0,000){\includegraphics[width=0.40\textwidth]{fig9c.eps}}
\put(210,330){(a)}
\put(210,215){(b)}
\put(210,100){(c)}
\end{picture}
\caption {Wavelet plane $w_F(a,b)$ calculated for different ratios
of signal-to-noise from $P(\lambda^2)$, obtained for the wide beam
(shown in Fig.~\protect{\ref{fig7}}b by the black line). From top to
bottom: signal without noise, signal-to-noise ratios of 2 and
0.5. The horizontal dashed line shows the upper bound of the domain
admissible to LOFAR-type observations. } \label{fig9}
\end{figure}

\section{Small-scale magnetic fields}
\label{turb}

In contrast to the traditional methods, wavelet-based RM Synthesis
opens a new option to quantify the small-scale component of magnetic
field (turbulent magnetic fields). Remember that it is widely
believed that one can isolate two contributions in total magnetic
field, i.e. the large-scale magnetic field and the small-scale one.
Correspondingly, the Faraday dispersion function $F$ and the Faraday
depth $\phi$ have two corresponding components, coming from
large-scale and turbulent fields. The traditional Burn (1966) theory
suggests to average $P$ over turbulent variations what yields in
additional depolarization effects like internal Faraday dispersion
and beam depolarization. In principle, one could try to modify
Eq.~(\ref{p_to_f}) to isolate the large-scale contribution and
parameterize the small-scale one by intrinsic Faraday dispersion and
beam depolarization. Wavelets allow us to perform this procedure in
a more straightforward way and to obtain more detailed information
concerning the small-scale component.

We start here with an example of a magnetic field which contains
large-scale and turbulent components (Fig.~\ref{fig7}a) and consider
a slab with thickness $2h = 1$\,kpc with a large-scale magnetic
field $B$ and a turbulent magnetic field with Kolmogorov spectrum
and r.m.s. value $b$ with the scale $l=0.1$\,kpc ($l/2h = 1/20$) and
$b/B = 2$. The polarized intensity obtained from such a slab is
affected by two depolarization effects. One effect known as internal
Faraday dispersion occurs because several (about 20 in our example)
independent turbulent cells are located at a given line of sight
passing through the slab. This effect occurs even for a very narrow
beam, with a cross-section with the slab surface containing just one
turbulent cell. The corresponding $P$ for two independent
realizations of the turbulent field is shown by dashed lines in
Fig.~\ref{fig7}b. The typical $P/P_{\rm max}$ ($P_{\rm max}$ is the
maximum value of $|P|$, which is achieved at lowest $\lambda^2$) is
***significant (about 10\%-20\%) in the range of $\lambda^2$=....***.

The other depolarization effect arises because the beam is usually
wide enough and contains many turbulent cells. If, say, the beam
diameter on the slab is 1\,kpc it contains about 100 independent
cells. The polarized emission passing through independent cells
contributes to $P$, leading to additional depolarization known as
beam depolarization. The $P$ calculated for a beam with a
cross-section containing 50 independent cells is shown by the solid
line in Fig.~\ref{fig7}b. We see that typical $P/P_{\rm max}$
***values are only a few percent for large $\lambda^2$.***

A usual intention here is to say that the radiation is almost
completely depolarized and useless for RM Synthesis. In contrast,
the wavelet method being applied to the $P$ shows a well-ordered
structure (Fig.~\ref{fig9}). The point is that wavelets separate
contributions from different spatial scales and associate them with
specific domains in the plot.

In Fig.~\ref{fig9}a, we see a contribution from the large-scale
magnetic field (the long horizontal feature at the upper part of the
panel) and various small-scale details at the bottom which represent
the contributions of individual turbulent cells. Having a limited
spectral range covered by the observations, we can obtain wavelet
coefficients in a limited part of the wavelet plane only. The border
of this part is given by Eq.~(\ref{crit}) which reads as
%
\begin{equation}
a_{\rm max} = \lambda_{\rm min}^{-2}.
 \label{acrit}
 \end{equation}
%
Here we take into account that for structures of size $a$ the range
of Faraday depth is of order $\Delta \phi = a^{-1}$. The upper limit
of the domain admissible for LOFAR observations is shown in
Fig.~\ref{fig9} by a horizontal dashed line. We conclude that the
contribution of the large-scale magnetic field remains inaccessible
for this type of observations while the small-scale details remain
visible. Note that the wavelet transform presented in the figure
allows to recognize the range of Faraday depths in which small-scale
structures occur (from 0 rad/m$^2$ to 38 rad/m$^2$ for the given
example). Obviously, using data at large $\lambda^2$ only we cannot
distinguish the turbulent magnetic field produced by tangling a
large-scale magnetic field \footnote{But a tangled field is NOT a
turbulent field - it may only look like! --Rainer} from a magnetic
field generated by a pure small-scale turbulent dynamo in absence of
large-scale magnetic field.

\begin{figure}
\begin{picture}(240,450)(0,0)
\put(0,335){\includegraphics[width=0.40\textwidth]{fig10a.eps}}
\put(0,220){\includegraphics[width=0.40\textwidth]{fig10b.eps}}
\put(0,110){\includegraphics[width=0.40\textwidth]{fig10c.eps}}
\put(0,000){\includegraphics[width=0.40\textwidth]{fig10d.eps}}
\put(210,435){(a)}
\put(210,320){(b)}
\put(210,210){(c)}
\put(210,100){(d)}
\end{picture}
\caption { Wavelet plane $w_F(a,b)$ calculated from the polarized
radio emission $P(\lambda^2)$ generated by a ``turbulent'' galactic
magnetic field superposed by a point source behind the galaxy (upper
panel a) and results of RM Synthesis: b - wavelet reconstruction
using the whole wavelet plane; c - wavelet reconstruction using only
the domains, marked in the wavelet plane by dashed lines; d -
reconstruction by standard RM Synthesis (the red line is for the
real part and the blue one for the imaginary part). }
\label{fig_last}\end{figure}

The method under discussion presumes that the signal-to-noise ratio
is sufficiently large. The lower panels of Fig.~\ref{fig9}b present
the result of the wavelet RM Synthesis for $P$ to which random
instrumental noise (independent in each spectral channel from the
noise in the other channels) is added. The signal-to-noise ratio
(calculated for the observational range $1.25 < \lambda < 2.5$\, m)
is about 2 (panel b) and the contribution of the small-scale fields
is easily distinguishable from that one which results from
instrumental noise. A visual inspection of these plots shows that
the signal-to-noise ratio about 2 or higher is sufficient to isolate
the range in Faraday space responsible for the small-scale magnetic
fields. For illustrative purposes we give an example with the
signal-to-noise ratio of about 0.5 and demonstrate that the
contribution of small-scale field cannot be distinguished from
instrumental noise (Fig.~\ref{fig9}c).


\section{A synthetic Example}
\label{SE}

To summarize our findings and to conclude what can be learnt from
polarized radio observations of galactic magnetic fields using
different RM Synthesis techniques we now consider a complex test
example. Suppose that we observe a galaxy which hosts both regular
and turbulent magnetic field (like the example considered in Section
\ref{turb}) and a strong point-like source of polarized radio
emission behind the galaxy on the same line of sight. Independent of
its position in physical space (coordinate $z$), this source in
Faraday space looks like attached to the galactic backside. For sake
of definiteness we suppose that the contribution to $F$ which comes
from the galaxy is purely real and that one from the point source is
purely imaginary. The task for the analysis is to restore the
intensity and the polarization angle of the point source, to
separate the contribution of large-scale and small-scale galactic
magnetic field and recognize the shape of the large-scale structure.

Fig.~\ref{fig_last}a shows the wavelet plane calculated for this
signal. We recognize here an extended structure in the regime of
large scales, $a \approx 20$ rad m$^{-2}$, which we identify with
the contribution of a galactic disc as a whole, small-scale details
at $a < 1$ rad m$^{-2}$, which we identify with the contributions of
the small-scale magnetic fields, and a bright horn located at $b
\approx 37$ rad m$^{-2}$. Then we perform the wavelet-based RM
Synthesis using the symmetry arguments to the galactic contribution
and considering the domain of the horn as a point-like source, i.e.
apply the symmetry condition again. The dashed lines in
Fig.~\ref{fig_last}a shows the domains in which the symmetry
arguments are used for calculation of $w_{-}$. Calculating $w_{-}$
we take into account the range of $b$, which are ***separated by***
not more than one scale in $a$ from the corresponding maximum of the
wavelet transform. Applying the symmetry arguments for the whole
domain would amplify the small-scale noise and reflect it with
respect to the position of the maximum.

The result of the reconstruction is shown at Fig.\ref{fig_last}b. We
can see a point-like source and a smooth contribution of the
large-scale magnetic field superimposed by fluctuations associated
with the small-scale fields. If we are not interested in the
small-scale details we ignore the wavelet coefficients, which are
remote from the corresponding maxima by more than one $a$ and obtain
a smoothed $F$, which represents the contribution of the mean field
in the disc and the point-like source only (Fig.~\ref{fig_last}c).
We see a smoothed real contribution from the mean field and an
imaginary point-like source, i.e. the method reproduces correctly
the polarization angles. For comparison in Fig.~\ref{fig_last}d
shows the result obtained by the standard RM technique (the phase
information is completely lost there).

\section{Discussion and Conclusions}
\label{DC}

We considered how to apply RM Synthesis to extract information
concerning magnetic fields of spiral galaxies from the polarized
emission data. The main attention was given to two related novelties
in the method, i.e. the symmetry argument and the wavelet technique.
Symmetry arguments open a possibility to reconstruct $P(\lambda^2)$
for negative $\lambda^2$ from $P(\lambda^2)$ for positive
$\lambda^2$. If $F$ contains several contributions (say, the disc
and an unresolved source) the reconstruction can be performed
locally in Faraday depth space using wavelets. The wavelet technique
appears to be useful in RM Synthesis in several points which are not
directly related to the problem of negative $\lambda^2$. In
particular, they allow to perform RM Synthesis locally in the
Faraday depth space and help to understand better what limitation is
imposed by the finite observational range $\lambda_{\rm min} <
\lambda < \lambda_{\rm max}$.

The idea of the RM Synthesis was applied previously mainly to the
sources which are point-like in Faraday space
\citep{2000A&A...356L..13H,2003A&A...403.1031H,2005A&A...441..931D,2009A&A...503..409H}.
We demonstrated that the traditional RM Synthesis for a point-like
source indirectly exploits a symmetry argument and, in this sense,
can be considered as a particular case of the method under
discussion.

Investigating the applications of RM Synthesis to the polarization
details associated with small-scale magnetic fields we isolated an
option which was not covered by \cite{Burn1966MNRAS.133...67B}
ideas. \cite{Burn1966MNRAS.133...67B} (see also
\cite{1998MNRAS.299..189S}) describes a contribution of small-scale
field in terms of Faraday dispersion and beam depolarization, i.e.
using averages over a corresponding statistical ensemble. Here we
exploit the complex polarization $P$ for RM Synthesis without any
averaging and demonstrate that it allows to obtain a range in
Faraday space where the contribution from small-scale field is
located.

A general conclusion concerning the applicability of RM Synthesis to
the interpretation of the radio polarization data for extended
sources like spiral galaxies is that quite severe requirements are
needed to provide the full applicability of the method.
\cite{2009IAUS..259..669B} show that RM Synthesis being applied at
the border of its applicability (say, if the number of frequency
channels is very small) can sometimes be misleading.

Traditional methods of the pattern recognition for magnetic
structures in spiral galaxies (e.g.
\cite{1990A&A...230..284R,1992A&A...264..396S,1997A&A...318..700B,
2004A&A...414...53F,2006A&A...458..441P,2008A&A...480...45S}) were
mainly based on the interpretation of polarization angles at a few
wavelengths only and exploited other information from $P$ in a very
limited wavelength range (cf. \cite{1998MNRAS.299..189S}). The
efforts of the above papers were concentrated mainly on the
distribution of polarization angles in the galactic image. The
modern wide-band observations with many frequency channels and RM
Synthesis open wide perspectives to extract much more information
from the polarization signal coming from a given point of the
galactic image, however, cannot fully replace the traditional
analysis of the distribution of the polarization patterns.
Obviously, the development of RM Synthesis in order to deal with
galactic images as a whole rather with just one point is a goal for
further developments, and wavelets are likely to be helpful here.
However, this is out of the scope of this paper.

\label{lastpage}

\section*{Acknowledgments} This work was supported by the
DFG-RFBR grant 08-02-92881. We thank George Heald for providing
observational data and useful discussions.

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\end{document}
